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Lambda theories allowing terms with a finite number of fixed points

Published online by Cambridge University Press:  29 July 2015

BENEDETTO INTRIGILA  and
RICHARD STATMAN
BENEDETTO INTRIGILA
Affiliation:
Dipartimento di Ingegneria dell'Impresa, Università di Roma ‘Tor Vergata,’Rome, Italy
RICHARD STATMAN
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA
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Abstract

A natural question in the λ-calculus asks what is the possible number of fixed points of a combinator (closed term). A complete answer to this question is still missing (Problem 25 of TLCA Open Problems List) and we investigate the related question about the number of fixed points of a combinator in λ-theories. We show the existence of a recursively enumerable lambda theory where the number is always one or infinite. We also show that there are λ-theories such that some terms have only two fixed points. In a first example, this is obtained by means of a non-constructive (more precisely non-r.e.) λ-theory where the range property is violated. A second, more complex example of a non-r.e. λ-theory (with a higher unsolvability degree) shows that some terms can have only two fixed points while the range property holds for every term.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Issue 3 , March 2017 , pp. 405 - 427
Copyright
Copyright © Cambridge University Press 2015 

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